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Electronic Journal: Southwest Journal of Pure and Applied Mathematics

Internet: http://rattler.cameron.edu/swjpam.html

ISSN 1083-0464

Issue 2, December 2003, pp. 9–17.

Submitted: October 13, 2003. Published: December 31 2003.

A SOLUTION TO AN ”UNSOLVED

PROBLEM IN NUMBER THEORY”

Allan J. MacLeod

Abstract. We discuss the problem of finding integer-sided triangles with the ratio base/altitude or altitude/base an integer. This problem is mentioned in Richard Guy’s book ”Unsolved Problems in Number Theory”. The problem is shown to be equivalent to finding rational points on a family of elliptic curves. Various computa-tional resources are used to find those integers in [1,99] which do appear, and also find the sides of example triangles.

A.M.S. (MOS) Subject Classification Codes.

11D25 , 11Y50

Key Words and Phrases.

Triangle, Elliptic curve, Rank, Descent

Department of Mathematics and Statistics, University of Paisley, High St., Paisley, Scotland. PA1 2BE

E-mail Address:[email protected] c

2003 Cameron University

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